How do I eliminate the element 't' from two given parametric equations?

Question

I'm stuck with this, and would appreciate if someone could guide me through it as I can't really take the exercise further without this knowledge.

I have two parametric equations (they represent the parametric equations of a points movement):

x = $e^{5t-1} + t^2$

y = $3t -1$

How do I remove the t element from these equations? In order to get the trajectory equation, I need to eliminate the 't' elements from both.

I need to reach a result that looks like this:

=>

Answer 1

I'm not sure if this will be satisfactory explanation.

From the second equation you get $t={y+1\over 3}$. Now put this in the first equation and you will have an equation without $t$.

Answer 2

The equation for $x$ is not linear, so there is no way that you can put it in the desired form.

You can get a locally linear approximation by looking at the tangents, but that is the best that you can do.

Answer 3

Solve $y=3t-1$ for $t$, that is $t=\frac {y+1}{3}$.

Plug this result for $t$ in the

$x= e^{5t-1} + t^2$

to get rid of $t$

If you are looking for an equation like the example that you have provided, you are not going to get it because first of all your example has t in every coordinate and it is linear while your problem in nonlinear.

Answer 4

As said in comments and answer, you cannot do much.

For a given $t$, you have $y=3t-1$; so you know $y$ for $t$; now, using $t=\frac 13 (y+1)$ and replacing in the expression of $x$, you then have $$x=\frac{1}{9} (y+1)^2+e^{\frac{5 y+2}{3}}$$ and nox, you know $x$.